Series

What they is...¶

The concept of a series is tightly coupled to sequences. They are simply the summation of all of the terms of sequence.

\[ \begin{align*} a_n = \{n\} && \text{a sequence} \\ \sum_{n=1}^\infty a_n && \text{a series} \end{align*} \]

Properties¶

Convergence¶

The idea of convergence foxr a series is different notion than convergence with a sequence. The idea is best examined through the idea of partial sums. If a series is the sum of all of the sequence terms, then a partial sum is the sum of the first N terms $ S_N $. We can begin with the first partial sum, the second partial sum, the third partial sum, and we can take that to the limit of infinity.

\[ \begin{align} &S = \sum_{n=1}^\infty a_n && \\ &S_n = \sum_{n=1}^N a_n && \\ &\{ S_n \}^\infty_{N=1} && \\ &S = L = \lim_{n \to \infty} S_n \end{align} \]

a series
a partial sum. The sum of the first N terms of $ { a_n }$
a sequence of partial sums where each $ S_n $ is the sum of the first n terms of the original sequence $ { a_n }$
the sum is the limit from n to $ \infty $ of the sequence of partial sums $ { S_n } $. In #2, $ S_n $ is a partial sum, but when written here, we are referring to the previous sentences understanding.

Imagine that we have pebbles and that we add them one by one to a scale and record the weight as we add each one (the $ S_N $ value). If as we record the weight it settles to a number, then we say that the series converges. (analogy idea from ChatGPT)

Absolute Convergence¶

A series that is the sum of all of the absolute terms of its sequence ( $ \sum | a_n | $ ), and is convergent, is so called absolutely convergent.

If a series is absolutely convergent, then it is also convergent.

Conditional Convergence¶

If the series is convergent, but the absolute series is not, then it is called conditionally convergent.

Tests¶

Divergency Test¶

It is impossible for a series to converge if the sequence of the terms does not approach 0. For example, if it approaches 1, then every term we continue to add in the series will continue to increase and the limit of the series won't settle down.

\[ \lim\limits_{n \to \infty} a_n \neq 0 \rightarrow \sum a_n \text{ will diverge} \]

The ratio test, root test, integral test could be used as well to show divergence.

Application¶

Take the limit of the sequence.

Integral Test¶

Show decreasing.

Comparison Test¶

What's bigger than it.

Limit Comparison Test¶

Show decreasing.

Alternating Series Test¶

Ratio Test¶

Tend to be used when we see factorials or things to the power of n Shows absolute convegence.

Root Test¶

Shows absolute convegence.

General Test Strategy¶

Check for divergence¶

For a series to converge, the terms of the sequence must approach zero; otherwise, the series will continue to grow and will be divergent. To check for this we will take the limit of the sequence and see if it is zero. If it is, then it's possible that the series may converge, if it does not equal zero then the series definietely diverges.

Check for type of series¶

Harmonic series¶

We know harmonic series are divergent.

Hyperharmonic (p-series)¶

We know hyperharmonic or p-series are convergent.

Geometric series¶

Geometric series will converge under special conditions. A geometric series $\sum_{n=0}^{\infty} ar^n$ will:

Converge if the common ratio, $|r| < 1$. The sum of the series in this case is given by:
- $ S = \frac{a}{1 - r} $ where $S$ is the sum of the series, $a$ is the first term, and $r$ is the common ratio.
Diverge if the common ratio, $|r| \geq 1$. This includes:
- $r = 1$, where the series becomes $\sum_{n=0}^{\infty} a$ which clearly diverges as it sums to infinity.
- $r > 1$ or $r \leq -1$, where the terms of the series grow without bound in absolute value.